Parameter estimation for bivariate shock models with singular distribution for censored data with concomitant order statistics

When two-component parallel systems are tested, the data consist of Type-Il censored data X-(i), i = 1,..., n, from one component, and their concomitants Y-[i] randomly censored at X-(r), the stopping time of the experiment. Marshall & Olkin's (1967) bivariate exponential distribution is used to illustrate statistical inference procedures developed for this data type. Although this data type is motivated practically, the likelihood is complicated, and maximum likelihood estimation is difficult, especially in the case where the parameter space is a non-open set. An iterative algorithm is proposed for finding maximum likelihood estimates. This article derives several properties of the maximum likelihood estimator (MLE) including existence, uniqueness, strong consistency and asymptotic distribution. It also develops an alternative estimation method with closed-form expressions based on marginal distributions, and derives its asymptotic properties. Compared with variances of the MLEs in the finite and large sample situations, the alternative estimator performs very well, especially when the correlation between X and Y is small.